196 MR JOHN DOUGALL ON AN ANALYTICAL THEORY OF 
form a complete infinite plate. By the general existence theorem of the subject, there 
exist values of u,v, w in the space within an internal edge, continuous at the edge with 
the values of the displacements of the original solid, and produced by edge tractions 
alone. Similarly, if we take any surface 8”, within the infinite plate, but completely 
enclosing the edge 8, there exist values of u,v, w continuous with the original u, v, w 
at the external edge, and becoming zero on 8”; these also being produced by edge trac- 
tions only, namely, on S and 8”. | 
If, then, we take w, v, w to be zero outside 8”, we obtain altogether a system of dis- — 
placements continuous throughout the infinite solid. The forces required to maintain 
this system are given directly by the general equations of equilibrium. These forces 
form areal distributions on 8, 8’, 8”, and are measured by the discontinuity of stress at 
these surfaces. Further, on the whole they make up an equilibrating system. But we 
have shown in the preceding pages how to find a solution for such a system of force, 
this solution giving displacements of order log R at most, and stresses of order R~ at 
most, at a great distance. Only one solution fulfilling these conditions being possible, — 
our solution is the solution. } 
Hence, finally, any displacement of a finite plate under edge tractions only is of the 
same form as that given by our infinite solid solutions for a certain system of areal 
force, distributed partly over the edges, and partly over an arbitrary external surface. — 
This is what we proposed to prove. 
35. General solution for an infinite solid under any forces. 
It is now easy to determine the most general form of displacement of an infinite 
solid, under null body force and face traction, and free from singularity at a finite 
distance. For if uw, v, w be any such displacement, then within any surface 8, however 
distant, we have proved that uw, v, w are given by the absolutely convergent series 
(98) zs aq lO): 
If we take a nght circular cylinder for the surface 8, the functions F which satisfy 
2 2 
equations of the form = ia +«’F =0 can be expressed in series of the form 
= JImkp(Am cos mw + Bm sin mo), 
and the only restriction on the coefticients A,,, B,, is that they must make the double — 
series in which the complete solution is thus expressed absolutely convergent for all 
values of p, however great. . 
The most general solution for any system of force applied at a finite distance is of 
course obtained by adding to this complete free solution the particular solution already 
investigated. It may be observed that this final result might have been obtained im — 
one step by the process of § 33, if in that article we had taken for uw, v, w any displace-_ 
ments under given body force and surface traction, instead of under edge traction only. 
The identity of the results of the two methods will be seen to depend essentially on the 
